Investigating Second Eigenvalue
Investigating Second Eigenvalue
if (require("PageRank")) {
library(PageRank)
}else{
devtools::install_github("ryangreenup/PageRank")
library(PageRank)
}Loading required package: PageRank library(pacman)
pacman::p_load(PageRank, devtools, Matrix, igraph, mise, tidyverse, rgl, latex2exp)
# mise()Looking at Density
Constants
Define some constants
m <- seq(from = 1, to = 10, length.out = 5)
m <- 5
beta <- seq(from = 1, to = 20, length.out = 10)
sz <- seq(from = 10, to = 1000, length.out = 10)
input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_varFunction to Build Graph
random_graph <- function(m, beta, size) {
g1 <- igraph::sample_pa(n = size, power = 3, m = m)
A <- igraph::get.adjacency(g1) # Row to column
A <- Matrix::t(A)
A_dens <- mean(A)
T <- PageRank::power_walk_prob_trans(A, beta = beta)
tr <- sum(diag(T))
e2 <- eigen(T, only.values = TRUE)$values[2] # R orders by descending magnitude
return(c(abs(e2), mean(A), tr))
}Return results
Map the function
rm(Y, data)object 'Y' not foundobject 'data' not foundnc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
X <- as.vector(input_var[i,])
Y[i,] <- random_graph(X$m, X$beta, X$size)
print(i/nrow(input_var))
}[1] 0.01
[1] 0.02
[1] 0.03
[1] 0.04
[1] 0.05
[1] 0.06
[1] 0.07
[1] 0.08
[1] 0.09
[1] 0.1
[1] 0.11
[1] 0.12
[1] 0.13
[1] 0.14
[1] 0.15
[1] 0.16
[1] 0.17
[1] 0.18
[1] 0.19
[1] 0.2
[1] 0.21
[1] 0.22
[1] 0.23
[1] 0.24
[1] 0.25
[1] 0.26
[1] 0.27
[1] 0.28
[1] 0.29
[1] 0.3
[1] 0.31
[1] 0.32
[1] 0.33
[1] 0.34
[1] 0.35
[1] 0.36
[1] 0.37
[1] 0.38
[1] 0.39
[1] 0.4
[1] 0.41
[1] 0.42
[1] 0.43
[1] 0.44
[1] 0.45
[1] 0.46
[1] 0.47
[1] 0.48
[1] 0.49
[1] 0.5
[1] 0.51
[1] 0.52
[1] 0.53
[1] 0.54
[1] 0.55
[1] 0.56
[1] 0.57
[1] 0.58
[1] 0.59
[1] 0.6
[1] 0.61
[1] 0.62
[1] 0.63
[1] 0.64
[1] 0.65
[1] 0.66
[1] 0.67
[1] 0.68
[1] 0.69
[1] 0.7
[1] 0.71
[1] 0.72
[1] 0.73
[1] 0.74
[1] 0.75
[1] 0.76
[1] 0.77
[1] 0.78
[1] 0.79
[1] 0.8
[1] 0.81[1] 0.82Plot Results
pairs(data)
cor(data)the standard deviation is zero m beta size eigenvalue2 A_dens trace
m 1 NA NA NA NA NA
beta NA 1.000000e+00 4.635990e-21 0.3038008 -2.642753e-21 -0.3880604
size NA 4.635990e-21 1.000000e+00 -0.6178130 -5.916467e-01 0.6586858
eigenvalue2 NA 3.038008e-01 -6.178130e-01 1.0000000 8.662592e-01 -0.9868650
A_dens NA -2.642753e-21 -5.916467e-01 0.8662592 1.000000e+00 -0.8183481
trace NA -3.880604e-01 6.586858e-01 -0.9868650 -8.183481e-01 1.0000000library(corrplot)corrplot 0.84 loadedcormat = cor(data, method = 'spearman')the standard deviation is zerocorrplot(cormat, method = "ellipse", type = "lower")
names(data)[1] "m" "beta" "size" "eigenvalue2" "A_dens"
[6] "trace" Let’s look at a 3d Output
names(data)[1] "m" "beta" "size" "eigenvalue2" "A_dens"
[6] "trace" library(plotly)
Attaching package: ‘plotly’
The following object is masked from ‘package:latex2exp’:
TeX
The following object is masked from ‘package:ggplot2’:
last_plot
The following object is masked from ‘package:igraph’:
groups
The following object is masked from ‘package:stats’:
filter
The following object is masked from ‘package:graphics’:
layout# d <- data[sample(1:nrow(data), 100),]
fig <- plot_ly(d, x = ~A_dens, y = ~beta, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'size'),
yaxis = list(title = 'Beta'),
zaxis = list(title = 'E2')))
fig`arrange_()` is deprecated as of dplyr 0.7.0.
Please use `arrange()` instead.
See vignette('programming') for more help
This warning is displayed once every 8 hours.
Call `lifecycle::last_warnings()` to see where this warning was generated.
names(data)[1] "m" "beta" "size" "eigenvalue2" "A_dens"
[6] "trace" library(plotly)
# d <- data[sample(1:nrow(data), 1000),]
fig <- plot_ly(d, x = ~A_dens, y = ~trace, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'Density'),
yaxis = list(title = 'trace'),
zaxis = list(title = 'E2')))
fig
NAClearly I should be able to model this.
Let’s look at density, it’s continuous so I should be able to take splices of A_dens.
A_dens is dependent on p, so I’ll just use different ps
m <- seq(from = 1, to = 9, length.out = 2)
beta <- seq(from = 1, to = 6, length.out = 10)
sz <- seq(from = 100, to = 1000, length.out = 4) %>% rev() # Big numbers first
input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var
rm(Y, data)object 'data' not foundnc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
X <- as.vector(input_var[i,])
Y[i,] <- random_graph(X$m, X$beta, X$size)
print(i/nrow(input_var))
}[1] 0.0125
[1] 0.025
[1] 0.0375
[1] 0.05
[1] 0.0625
[1] 0.075
[1] 0.0875
[1] 0.1
[1] 0.1125
[1] 0.125
[1] 0.1375
[1] 0.15
[1] 0.1625
[1] 0.175
[1] 0.1875
[1] 0.2
[1] 0.2125
[1] 0.225
[1] 0.2375
[1] 0.25
[1] 0.2625
[1] 0.275
[1] 0.2875
[1] 0.3
[1] 0.3125
[1] 0.325
[1] 0.3375
[1] 0.35
[1] 0.3625
[1] 0.375
[1] 0.3875
[1] 0.4
[1] 0.4125
[1] 0.425
[1] 0.4375
[1] 0.45
[1] 0.4625
[1] 0.475
[1] 0.4875
[1] 0.5
[1] 0.5125
[1] 0.525
[1] 0.5375
[1] 0.55
[1] 0.5625
[1] 0.575
[1] 0.5875
[1] 0.6
[1] 0.6125
[1] 0.625
[1] 0.6375
[1] 0.65
[1] 0.6625
[1] 0.675
[1] 0.6875
[1] 0.7
[1] 0.7125
[1] 0.725
[1] 0.7375
[1] 0.75
[1] 0.7625
[1] 0.775
[1] 0.7875
[1] 0.8
[1] 0.8125
[1] 0.825
[1] 0.8375
[1] 0.85
[1] 0.8625
[1] 0.875
[1] 0.8875
[1] 0.9
[1] 0.9125
[1] 0.925
[1] 0.9375
[1] 0.95
[1] 0.9625
[1] 0.975
[1] 0.9875
[1] 1if (sum(abs(Y) != abs(Re(Y))) == 0) {
Y <- Re(Y)
}
nrow(input_var)[1] 80nrow(Y)[1] 80Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data2 <- cbind(input_var, Y)) %>% head()# save(data2, file = "/home/ryan/Sync/Studies/2020Spring/DiscProj/data2_ba.rdata")
# load(file = "/home/ryan/Sync/Studies/2020Spring/DiscProj/data2_ba.rdata")data2$p <- (data2$m/data2$size)
names(data2)[1] "m" "beta" "size" "eigenvalue2" "A_dens"
[6] "trace" "p" ggplot(data2, mapping = aes(col = factor(p), x = beta, y = eigenvalue2)) +
geom_point(size = 0.5) +
stat_smooth(method = 'lm') +
scale_size_continuous(range = c(0.1,1)) +
labs(x = "Beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
guides(col = guide_legend("Link Density (by m/n)")) +
theme_bw()
It appears that m/n=density is its own feature, it does not affect E2 linearly so lets hold beta constant and see how it affects E2.
If for instance m/n increases, say quadratically for a constant beta, then I should try and fit a linear model to (m/n)^2*beta
load(file = "/tmp/a.rdata")
cannot open compressed file '/tmp/a.rdata', probable reason 'No such file or directory'Error in readChar(con, 5L, useBytes = TRUE) : cannot open the connectiondata2$p <- (data2$m/data2$size)
names(data2)[1] "m" "beta" "size" "eigenvalue2" "A_dens"
[6] "trace" "p" ggplot(data2, mapping = aes(col = factor(size), x = m/size, y = eigenvalue2)) +
geom_point(size = 0.5) +
stat_smooth(method = 'lm') +
scale_size_continuous(range = c(0.1,1)) +
labs(x = "m/size", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
guides(col = guide_legend("Size")) +
theme_bw()
So what about when I very m, i.e. lets map that to colour now:
Even though it’s hard to capture the behaviour of m/n, the size of the graph and the average outdegree are going to be constants and so this relationship should persist for graphs where m<<n.
actually if they are all linearly related, maybe I could use a multiple linear regression?
m <- 1:8
beta <- seq(from = 1, to = 20, length.out = 100)
sz <- seq(from = 100, to = 1000, length.out = 20) %>% rev() # Big numbers first
input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var
rm(Y, data)object 'data' not foundnc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
X <- as.vector(input_var[i,])
Y[i,] <- random_graph(X$m, X$beta, X$size)
print(i/nrow(input_var))
}[1] 6.25e-05
[1] 0.000125
[1] 0.0001875
[1] 0.00025
[1] 0.0003125
[1] 0.000375
[1] 0.0004375
[1] 5e-04
[1] 0.0005625
[1] 0.000625
[1] 0.0006875
[1] 0.00075
[1] 0.0008125
[1] 0.000875
[1] 0.0009375
[1] 0.001
[1] 0.0010625
[1] 0.001125
[1] 0.0011875
[1] 0.00125
[1] 0.0013125
[1] 0.001375
[1] 0.0014375
[1] 0.0015
[1] 0.0015625
[1] 0.001625
[1] 0.0016875
[1] 0.00175
[1] 0.0018125
[1] 0.001875
[1] 0.0019375
[1] 0.002
[1] 0.0020625
[1] 0.002125
[1] 0.0021875
[1] 0.00225
[1] 0.0023125
[1] 0.002375
[1] 0.0024375
[1] 0.0025
[1] 0.0025625
[1] 0.002625
[1] 0.0026875
[1] 0.00275
[1] 0.0028125
[1] 0.002875
[1] 0.0029375
[1] 0.003
[1] 0.0030625
[1] 0.003125
[1] 0.0031875
[1] 0.00325
[1] 0.0033125
[1] 0.003375
[1] 0.0034375
[1] 0.0035
[1] 0.0035625
[1] 0.003625
[1] 0.0036875
[1] 0.00375
[1] 0.0038125
[1] 0.003875
[1] 0.0039375
[1] 0.004
[1] 0.0040625
[1] 0.004125
[1] 0.0041875
[1] 0.00425
[1] 0.0043125
[1] 0.004375
[1] 0.0044375
[1] 0.0045
[1] 0.0045625
[1] 0.004625
[1] 0.0046875
[1] 0.00475
[1] 0.0048125
[1] 0.004875
[1] 0.0049375
[1] 0.005
[1] 0.0050625
[1] 0.005125
[1] 0.0051875
[1] 0.00525
[1] 0.0053125
[1] 0.005375
[1] 0.0054375
[1] 0.0055
[1] 0.0055625
[1] 0.005625
[1] 0.0056875
[1] 0.00575
[1] 0.0058125
[1] 0.005875
[1] 0.0059375
[1] 0.006
[1] 0.0060625
[1] 0.006125
[1] 0.0061875
[1] 0.00625
[1] 0.0063125
[1] 0.006375
[1] 0.0064375
[1] 0.0065
[1] 0.0065625
[1] 0.006625
[1] 0.0066875
[1] 0.00675
[1] 0.0068125
[1] 0.006875
[1] 0.0069375
[1] 0.007
[1] 0.0070625
[1] 0.007125
[1] 0.0071875
[1] 0.00725
[1] 0.0073125
[1] 0.007375
[1] 0.0074375
[1] 0.0075
[1] 0.0075625
[1] 0.007625
[1] 0.0076875
[1] 0.00775
[1] 0.0078125
[1] 0.007875
[1] 0.0079375
[1] 0.008
[1] 0.0080625
[1] 0.008125
[1] 0.0081875
[1] 0.00825
[1] 0.0083125
[1] 0.008375
[1] 0.0084375
[1] 0.0085
[1] 0.0085625
[1] 0.008625
[1] 0.0086875
[1] 0.00875
[1] 0.0088125
[1] 0.008875
[1] 0.0089375
[1] 0.009
[1] 0.0090625
[1] 0.009125
[1] 0.0091875
[1] 0.00925
[1] 0.0093125
[1] 0.009375
[1] 0.0094375
[1] 0.0095
[1] 0.0095625
[1] 0.009625
[1] 0.0096875
[1] 0.00975
[1] 0.0098125
[1] 0.009875
[1] 0.0099375
[1] 0.01
[1] 0.0100625
[1] 0.010125
[1] 0.0101875
[1] 0.01025
[1] 0.0103125
[1] 0.010375
[1] 0.0104375
[1] 0.0105
[1] 0.0105625
[1] 0.010625
[1] 0.0106875
[1] 0.01075
[1] 0.0108125
[1] 0.010875
[1] 0.0109375
[1] 0.011
[1] 0.0110625
[1] 0.011125
[1] 0.0111875
[1] 0.01125
[1] 0.0113125
[1] 0.011375
[1] 0.0114375
[1] 0.0115
[1] 0.0115625
[1] 0.011625
[1] 0.0116875
[1] 0.01175
[1] 0.0118125
[1] 0.011875
[1] 0.0119375
[1] 0.012
[1] 0.0120625
[1] 0.012125
[1] 0.0121875
[1] 0.01225
[1] 0.0123125
[1] 0.012375
[1] 0.0124375
[1] 0.0125
[1] 0.0125625
[1] 0.012625
[1] 0.0126875
[1] 0.01275
[1] 0.0128125
[1] 0.012875
[1] 0.0129375
[1] 0.013
[1] 0.0130625
[1] 0.013125
[1] 0.0131875
[1] 0.01325
[1] 0.0133125
[1] 0.013375
[1] 0.0134375
[1] 0.0135
[1] 0.0135625
[1] 0.013625
[1] 0.0136875
[1] 0.01375
[1] 0.0138125
[1] 0.013875
[1] 0.0139375
[1] 0.014
[1] 0.0140625
[1] 0.014125
[1] 0.0141875
[1] 0.01425
[1] 0.0143125
[1] 0.014375
[1] 0.0144375
[1] 0.0145
[1] 0.0145625
[1] 0.014625
[1] 0.0146875
[1] 0.01475
[1] 0.0148125
[1] 0.014875
[1] 0.0149375
[1] 0.015
[1] 0.0150625
[1] 0.015125
[1] 0.0151875
[1] 0.01525
[1] 0.0153125
[1] 0.015375
[1] 0.0154375
[1] 0.0155
[1] 0.0155625
[1] 0.015625
[1] 0.0156875
[1] 0.01575
[1] 0.0158125
[1] 0.015875
[1] 0.0159375
[1] 0.016
[1] 0.0160625
[1] 0.016125
[1] 0.0161875
[1] 0.01625
[1] 0.0163125
[1] 0.016375
[1] 0.0164375
[1] 0.0165
[1] 0.0165625
[1] 0.016625
[1] 0.0166875
[1] 0.01675
[1] 0.0168125
[1] 0.016875
[1] 0.0169375
[1] 0.017
[1] 0.0170625
[1] 0.017125
[1] 0.0171875
[1] 0.01725
[1] 0.0173125
[1] 0.017375
[1] 0.0174375
[1] 0.0175
[1] 0.0175625
[1] 0.017625
[1] 0.0176875
[1] 0.01775
[1] 0.0178125
[1] 0.017875
[1] 0.0179375
[1] 0.018
[1] 0.0180625
[1] 0.018125
[1] 0.0181875
[1] 0.01825
[1] 0.0183125
[1] 0.018375
[1] 0.0184375
[1] 0.0185
[1] 0.0185625
[1] 0.018625
[1] 0.0186875
[1] 0.01875
[1] 0.0188125
[1] 0.018875
[1] 0.0189375
[1] 0.019
[1] 0.0190625
[1] 0.019125
[1] 0.0191875
[1] 0.01925
[1] 0.0193125
[1] 0.019375
[1] 0.0194375
[1] 0.0195
[1] 0.0195625
[1] 0.019625
[1] 0.0196875
[1] 0.01975
[1] 0.0198125
[1] 0.019875
[1] 0.0199375
[1] 0.02
[1] 0.0200625
[1] 0.020125
[1] 0.0201875
[1] 0.02025
[1] 0.0203125
[1] 0.020375
[1] 0.0204375
[1] 0.0205
[1] 0.0205625
[1] 0.020625
[1] 0.0206875
[1] 0.02075
[1] 0.0208125
[1] 0.020875
[1] 0.0209375
[1] 0.021
[1] 0.0210625
[1] 0.021125
[1] 0.0211875
[1] 0.02125
[1] 0.0213125
[1] 0.021375
[1] 0.0214375
[1] 0.0215
[1] 0.0215625
[1] 0.021625
[1] 0.0216875
[1] 0.02175
[1] 0.0218125
[1] 0.021875
[1] 0.0219375
[1] 0.022
[1] 0.0220625
[1] 0.022125
[1] 0.0221875
[1] 0.02225
[1] 0.0223125
[1] 0.022375
[1] 0.0224375
[1] 0.0225
[1] 0.0225625
[1] 0.022625
[1] 0.0226875
[1] 0.02275
[1] 0.0228125
[1] 0.022875
[1] 0.0229375
[1] 0.023
[1] 0.0230625
[1] 0.023125
[1] 0.0231875
[1] 0.02325
[1] 0.0233125
[1] 0.023375
[1] 0.0234375
[1] 0.0235
[1] 0.0235625
[1] 0.023625
[1] 0.0236875
[1] 0.02375
[1] 0.0238125
[1] 0.023875
[1] 0.0239375
[1] 0.024
[1] 0.0240625
[1] 0.024125
[1] 0.0241875
[1] 0.02425
[1] 0.0243125
[1] 0.024375
[1] 0.0244375
[1] 0.0245
[1] 0.0245625
[1] 0.024625
[1] 0.0246875
[1] 0.02475
[1] 0.0248125
[1] 0.024875
[1] 0.0249375
[1] 0.025
[1] 0.0250625
[1] 0.025125
[1] 0.0251875
[1] 0.02525
[1] 0.0253125
[1] 0.025375
[1] 0.0254375
[1] 0.0255
[1] 0.0255625
[1] 0.025625
[1] 0.0256875
[1] 0.02575
[1] 0.0258125
[1] 0.025875
[1] 0.0259375
[1] 0.026
[1] 0.0260625
[1] 0.026125
[1] 0.0261875
[1] 0.02625
[1] 0.0263125
[1] 0.026375
[1] 0.0264375
[1] 0.0265
[1] 0.0265625
[1] 0.026625
[1] 0.0266875
[1] 0.02675
[1] 0.0268125
[1] 0.026875
[1] 0.0269375
[1] 0.027
[1] 0.0270625
[1] 0.027125
[1] 0.0271875
[1] 0.02725
[1] 0.0273125
[1] 0.027375
[1] 0.0274375
[1] 0.0275
[1] 0.0275625
[1] 0.027625
[1] 0.0276875
[1] 0.02775
[1] 0.0278125
[1] 0.027875
[1] 0.0279375
[1] 0.028
[1] 0.0280625
[1] 0.028125
[1] 0.0281875
[1] 0.02825
[1] 0.0283125
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[1] 0.98375
[1] 0.9838125
[1] 0.983875
[1] 0.9839375
[1] 0.984
[1] 0.9840625
[1] 0.984125
[1] 0.9841875
[1] 0.98425
[1] 0.9843125
[1] 0.984375
[1] 0.9844375
[1] 0.9845
[1] 0.9845625
[1] 0.984625
[1] 0.9846875
[1] 0.98475
[1] 0.9848125
[1] 0.984875
[1] 0.9849375
[1] 0.985
[1] 0.9850625
[1] 0.985125
[1] 0.9851875
[1] 0.98525
[1] 0.9853125
[1] 0.985375
[1] 0.9854375
[1] 0.9855
[1] 0.9855625
[1] 0.985625
[1] 0.9856875
[1] 0.98575
[1] 0.9858125
[1] 0.985875
[1] 0.9859375
[1] 0.986
[1] 0.9860625
[1] 0.986125
[1] 0.9861875
[1] 0.98625
[1] 0.9863125
[1] 0.986375
[1] 0.9864375
[1] 0.9865
[1] 0.9865625
[1] 0.986625
[1] 0.9866875
[1] 0.98675
[1] 0.9868125
[1] 0.986875
[1] 0.9869375
[1] 0.987
[1] 0.9870625
[1] 0.987125
[1] 0.9871875
[1] 0.98725
[1] 0.9873125
[1] 0.987375
[1] 0.9874375
[1] 0.9875
[1] 0.9875625
[1] 0.987625
[1] 0.9876875
[1] 0.98775
[1] 0.9878125
[1] 0.987875
[1] 0.9879375
[1] 0.988
[1] 0.9880625
[1] 0.988125
[1] 0.9881875
[1] 0.98825
[1] 0.9883125
[1] 0.988375
[1] 0.9884375
[1] 0.9885
[1] 0.9885625
[1] 0.988625
[1] 0.9886875
[1] 0.98875
[1] 0.9888125
[1] 0.988875
[1] 0.9889375
[1] 0.989
[1] 0.9890625
[1] 0.989125
[1] 0.9891875
[1] 0.98925
[1] 0.9893125
[1] 0.989375
[1] 0.9894375
[1] 0.9895
[1] 0.9895625
[1] 0.989625
[1] 0.9896875
[1] 0.98975
[1] 0.9898125
[1] 0.989875
[1] 0.9899375
[1] 0.99
[1] 0.9900625
[1] 0.990125
[1] 0.9901875
[1] 0.99025
[1] 0.9903125
[1] 0.990375
[1] 0.9904375
[1] 0.9905
[1] 0.9905625
[1] 0.990625
[1] 0.9906875
[1] 0.99075
[1] 0.9908125
[1] 0.990875
[1] 0.9909375
[1] 0.991
[1] 0.9910625
[1] 0.991125
[1] 0.9911875
[1] 0.99125
[1] 0.9913125
[1] 0.991375
[1] 0.9914375
[1] 0.9915
[1] 0.9915625
[1] 0.991625
[1] 0.9916875
[1] 0.99175
[1] 0.9918125
[1] 0.991875
[1] 0.9919375
[1] 0.992
[1] 0.9920625
[1] 0.992125
[1] 0.9921875
[1] 0.99225
[1] 0.9923125
[1] 0.992375
[1] 0.9924375
[1] 0.9925
[1] 0.9925625
[1] 0.992625
[1] 0.9926875
[1] 0.99275
[1] 0.9928125
[1] 0.992875
[1] 0.9929375
[1] 0.993
[1] 0.9930625
[1] 0.993125
[1] 0.9931875
[1] 0.99325
[1] 0.9933125
[1] 0.993375
[1] 0.9934375
[1] 0.9935
[1] 0.9935625
[1] 0.993625
[1] 0.9936875
[1] 0.99375
[1] 0.9938125
[1] 0.993875
[1] 0.9939375
[1] 0.994
[1] 0.9940625
[1] 0.994125
[1] 0.9941875
[1] 0.99425
[1] 0.9943125
[1] 0.994375
[1] 0.9944375
[1] 0.9945
[1] 0.9945625
[1] 0.994625
[1] 0.9946875
[1] 0.99475
[1] 0.9948125
[1] 0.994875
[1] 0.9949375
[1] 0.995
[1] 0.9950625
[1] 0.995125
[1] 0.9951875
[1] 0.99525
[1] 0.9953125
[1] 0.995375
[1] 0.9954375
[1] 0.9955
[1] 0.9955625
[1] 0.995625
[1] 0.9956875
[1] 0.99575
[1] 0.9958125
[1] 0.995875
[1] 0.9959375
[1] 0.996
[1] 0.9960625
[1] 0.996125
[1] 0.9961875
[1] 0.99625
[1] 0.9963125
[1] 0.996375
[1] 0.9964375
[1] 0.9965
[1] 0.9965625
[1] 0.996625
[1] 0.9966875
[1] 0.99675
[1] 0.9968125
[1] 0.996875
[1] 0.9969375
[1] 0.997
[1] 0.9970625
[1] 0.997125
[1] 0.9971875
[1] 0.99725
[1] 0.9973125
[1] 0.997375
[1] 0.9974375
[1] 0.9975
[1] 0.9975625
[1] 0.997625
[1] 0.9976875
[1] 0.99775
[1] 0.9978125
[1] 0.997875
[1] 0.9979375
[1] 0.998
[1] 0.9980625
[1] 0.998125
[1] 0.9981875
[1] 0.99825
[1] 0.9983125
[1] 0.998375
[1] 0.9984375
[1] 0.9985
[1] 0.9985625
[1] 0.998625
[1] 0.9986875
[1] 0.99875
[1] 0.9988125
[1] 0.998875
[1] 0.9989375
[1] 0.999
[1] 0.9990625
[1] 0.999125
[1] 0.9991875
[1] 0.99925
[1] 0.9993125
[1] 0.999375
[1] 0.9994375
[1] 0.9995
[1] 0.9995625
[1] 0.999625
[1] 0.9996875
[1] 0.99975
[1] 0.9998125
[1] 0.999875
[1] 0.9999375
[1] 1if (sum(abs(Y) != abs(Re(Y))) == 0) {
Y <- Re(Y)
}
nrow(input_var)[1] 16000nrow(Y)[1] 16000Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data2 <- cbind(input_var, Y)) %>% head()
mod <- lm(eigenvalue2 ~ m/size + size + m + beta, data2)
summary(mod)
Call:
lm(formula = eigenvalue2 ~ m/size + size + m + beta, data = data2)
Residuals:
Min 1Q Median 3Q Max
-0.054228 -0.011168 -0.004100 0.007102 0.116799
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.953e-03 7.478e-04 11.97 <2e-16 ***
m 6.435e-03 1.383e-04 46.53 <2e-16 ***
size -4.398e-05 1.137e-06 -38.67 <2e-16 ***
beta 3.207e-03 2.544e-05 126.06 <2e-16 ***
m:size -6.271e-06 2.252e-07 -27.85 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01783 on 15995 degrees of freedom
Multiple R-squared: 0.707, Adjusted R-squared: 0.707
F-statistic: 9650 on 4 and 15995 DF, p-value: < 2.2e-16plot(mod)
mod <- lm(eigenvalue2 ~ m/size + m + size + beta , data = data2)
max(data2$m/data2$size)[1] 0.08summary(mod)
Call:
lm(formula = eigenvalue2 ~ m/size + m + size + beta, data = data2)
Residuals:
Min 1Q Median 3Q Max
-0.054228 -0.011168 -0.004100 0.007102 0.116799
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.953e-03 7.478e-04 11.97 <2e-16 ***
m 6.435e-03 1.383e-04 46.53 <2e-16 ***
size -4.398e-05 1.137e-06 -38.67 <2e-16 ***
beta 3.207e-03 2.544e-05 126.06 <2e-16 ***
m:size -6.271e-06 2.252e-07 -27.85 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.01783 on 15995 degrees of freedom
Multiple R-squared: 0.707, Adjusted R-squared: 0.707
F-statistic: 9650 on 4 and 15995 DF, p-value: < 2.2e-16# layout(mat = matrix(1:4, nrow = 2))
# plot(mod)
# mean(mod$residuals**2)
hist(mod$residuals, breaks = 200)
Clearly non-normal residuals but +-5% is pretty good for a very complicated result.
I think I already looked at this but this again shows the relly clear, near linear relationship.
ggplot(data2, mapping = aes(col = factor(m), x = beta, y = eigenvalue2)) +
geom_smooth(size = 0.5) +
# stat_smooth() +
scale_size_continuous(range = c(0.1,1)) +
labs(x = "beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
guides(col = guide_legend("Vertices Added\n Each Step")) +
theme_bw()
Obviously we can’t know m, but, we can know the average out degree, so let’s, take a graph, measure the average out degree, use our model, measure it’s performance.
---
title: "Investigating Second Eigenvalue"
output: html_notebook
---

# Investigating Second Eigenvalue

```{r}
  if (require("PageRank")) {
      library(PageRank)
    }else{
      devtools::install_github("ryangreenup/PageRank")
      library(PageRank)
    }

  library(pacman)
  pacman::p_load(PageRank, devtools, Matrix, igraph, mise, tidyverse, rgl, latex2exp)
#  mise()
```



## Looking at Density

### Constants

Define some constants

```{r}
m    <- seq(from = 1, to = 10, length.out = 5)
m <- 5
beta <- seq(from = 1, to = 20, length.out = 10)
sz <- seq(from = 10, to = 1000, length.out = 10)
input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var
```

### Function to Build Graph

```{r}
random_graph <- function(m, beta, size) {
      g1 <- igraph::sample_pa(n = size, power = 3, m = m)
      A <- igraph::get.adjacency(g1) # Row to column
      A <- Matrix::t(A)

      A_dens <- mean(A)
      T      <- PageRank::power_walk_prob_trans(A, beta = beta)
      tr     <- sum(diag(T))
      e2     <- eigen(T, only.values = TRUE)$values[2] # R orders by descending magnitude
      return(c(abs(e2), mean(A), tr))
}
```

### Return results

Map the function

```{r}
rm(Y, data)
nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$m, X$beta, X$size)
  print(i/nrow(input_var))
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data <- cbind(input_var, Y)) %>% head()
```


### Plot Results


```{r}
pairs(data)
cor(data)
library(corrplot)
cormat = cor(data, method = 'spearman')
corrplot(cormat, method = "ellipse", type = "lower")
names(data)
```
Let's look at a 3d Output

```{r}
names(data)
library(plotly)

# d <- data[sample(1:nrow(data), 100),]

fig <- plot_ly(d, x = ~A_dens, y = ~beta, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'size'),
                     yaxis = list(title = 'Beta'),
                     zaxis = list(title = 'E2')))

fig

```

```{r}

names(data)
library(plotly)

# d <- data[sample(1:nrow(data), 1000),]

fig <- plot_ly(d, x = ~A_dens, y = ~trace, z = ~eigenvalue2)
fig <- fig %>% add_markers(size = 1)
fig <- fig %>% layout(scene = list(xaxis = list(title = 'Density'),
                     yaxis = list(title = 'trace'),
                     zaxis = list(title = 'E2')))

fig

```


Clearly I should be able to model this.

Let's look at density, it's continuous so I should be able to  take splices of A_dens.

A_dens is dependent on p, so I'll just use different ps


```{r}
m    <- seq(from = 1, to = 9, length.out = 2)
beta <- seq(from = 1, to = 6, length.out = 10)
sz <- seq(from = 100, to = 1000, length.out = 4) %>% rev() # Big numbers first 

input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var


rm(Y, data)
nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$m, X$beta, X$size)
  print(i/nrow(input_var))
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data2 <- cbind(input_var, Y)) %>% head()
# save(data2, file = "/home/ryan/Sync/Studies/2020Spring/DiscProj/data2_ba.rdata")
# load(file = "/home/ryan/Sync/Studies/2020Spring/DiscProj/data2_ba.rdata")
```


```{r}
data2$p <- (data2$m/data2$size)
names(data2)
ggplot(data2, mapping = aes(col = factor(p), x = beta, y = eigenvalue2)) +
  geom_point(size = 0.5) +
  stat_smooth(method = 'lm') +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "Beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Link Density (by m/n)"))  +
  theme_bw()
```
It appears that m/n=density is its own feature, it does not affect E2 linearly so lets hold beta constant and see how it affects E2.

If for instance m/n increases, say quadratically for a constant beta,
then I should try and fit a linear model to (m/n)^2*beta

```{r}
m    <- seq(from = 1, to = 9, length.out = 20)
beta <- seq(from = 1, to = 6, length.out = 10)
beta <- 25
sz <- seq(from = 100, to = 1000, length.out = 10) %>% rev() # Big numbers first 

input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var


rm(Y, data)
nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$m, X$beta, X$size)
  print(i/nrow(input_var))
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data2 <- cbind(input_var, Y)) %>% head()
```



```{r}
data2$p <- (data2$m/data2$size)
names(data2)
ggplot(data2, mapping = aes(col = factor(size), x = m/size, y = eigenvalue2)) +
  geom_point(size = 0.5) +
  stat_smooth(method = 'lm') +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "m/size", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Size"))  +
  theme_bw()
```
So what about when I very m, i.e. lets map that to colour now:

```{r}
m <- 1:8
beta <- seq(from = 1, to = 6, length.out = 10)
beta <- 5
sz <- seq(from = 100, to = 1000, length.out = 20) %>% rev() # Big numbers first 

input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var


rm(Y, data)
nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$m, X$beta, X$size)
  print(i/nrow(input_var))
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data2 <- cbind(input_var, Y)) %>% head()



data2$p <- (data2$m/data2$size)
names(data2)
ggplot(data2, mapping = aes(col = factor(m), x = m/size, y = eigenvalue2)) +
  geom_point(size = 0.5) +
  stat_smooth() +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "m/size", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Vertices Added\n Each Step"))  +
  theme_bw()
```


Even though it's hard to capture the behaviour of m/n, the size of the graph and the average outdegree are going to be constants and so this relationship should persist
for graphs where m<<n.



actually if they are all linearly related, maybe I could use a multiple linear regression?

```{r}
m <- 1:8
beta <- seq(from = 1, to = 20, length.out = 100)
sz <- seq(from = 100, to = 1000, length.out = 20) %>% rev() # Big numbers first 

input_var <- expand.grid("m" = m, "beta" = beta, "size" = sz)
input_var


rm(Y, data)
nc <- length(random_graph(1, 1, 1))
Y <- matrix(ncol = nc, nrow = nrow(input_var))
for (i in 1:nrow(input_var)) {
  X <- as.vector(input_var[i,])
  Y[i,] <-  random_graph(X$m, X$beta, X$size)
  print(i/nrow(input_var))
}
if (sum(abs(Y) != abs(Re(Y))) == 0) {
  Y <- Re(Y)
}
nrow(input_var)
nrow(Y)
Y <- as.data.frame(Y); colnames(Y) <- c("eigenvalue2", "A_dens", "trace")
(data2 <- cbind(input_var, Y)) %>% head()







mod <- lm(eigenvalue2 ~ m/size + size + m + beta, data2)
summary(mod)
plot(mod)
```





```{r}
mod <- lm(eigenvalue2 ~ m/size + m + size + beta , data = data2)
max(data2$m/data2$size)
summary(mod)
# layout(mat = matrix(1:4, nrow = 2))
# plot(mod)
# mean(mod$residuals**2)
hist(mod$residuals, breaks = 200)
```



Clearly non-normal residuals but +-5% is pretty good for a very complicated result.





I think I already looked at this but this again shows the relly clear, near linear relationship.


```{r}
ggplot(data2, mapping = aes(col = factor(m), x = beta, y = eigenvalue2)) +
  geom_smooth(size = 0.5) +
#  stat_smooth() +
  scale_size_continuous(range = c(0.1,1)) +
  labs(x = "beta", y = TeX("Second Eigenvalue"), title = TeX("Second Eigenvalue given Matrix Density") ) +
  guides(col = guide_legend("Vertices Added\n Each Step"))  +
  theme_bw()
```

Obviously we can't know m, but, we can know the average out degree, so let's,
take a graph,
measure the average out degree,
use our model,
measure it's performance.



```{r}

```



























